In mathematics, the syzygetic pencil or Hesse pencil, named for Otto Hesse, is a pencil (one-dimensional family) of cubic plane elliptic curves in the complex projective plane, defined by the equation Each curve in the family is determined by a pair of parameter values (not both zero) and consists of the points in the plane whose homogeneous coordinates satisfy the equation for those parameters. Multiplying both and by the same scalar does not change the curve, so there is only one degree of freedom in selecting a curve from the pencil, but the two-parameter form given above allows either or (but not both) to be set to zero. Each curve in the pencil passes through the nine points of the complex projective plane whose homogeneous coordinates are some permutation of 0, –1, and a cube root of unity. There are three roots of unity, and six permutations per root, giving 18 choices for the homogeneous coordinates of each point, but they are equivalent in pairs giving only nine points. The family of cubics through these nine points forms the Hesse pencil. More generally, one can replace the complex numbers by any field containing a cube root of unity and define the Hesse pencil over this field to be the family of cubics through these nine points. The nine common points of the Hesse pencil are the inflection points of each of the cubics in the pencil. Any line that passes through at least two of these nine points passes through exactly three of them; the nine points and twelve lines through triples of points form the Hesse configuration. Every elliptic curve is birationally equivalent to a curve of the Hesse pencil; this is the Hessian form of an elliptic curve. However, the parameters of the Hessian form may belong to an extension field of the field of definition of the original curve. (Wikipedia).
This shows a 3d printed mobile produced using shapeways.com. This is joint work with Marco Mahler. This is available at http://shpws.me/nPha.
From playlist 3D printing
This shows a 3d print of a mathematical sculpture I produced using shapeways.com. This model is available at http://shpws.me/2toQ.
From playlist 3D printing
This shows a 3d printed mobile produced using shapeways.com. This is joint work with Marco Mahler. This is available at http://shpws.me/nPh7.
From playlist 3D printing
How To Make a Hexaflexagon: The Definitive Guide
Printable patterns: http://vihart.com/hexaflexagons Shirts (we only printed a limited run, sorry if we're sold out of your favourite!): https://store.dftba.com/collections/vi-hart This video is supported not mostly by shirts but mostly by viewers like you! Thanks especially to: Caleb Wri
From playlist Hexaflexagon Series
Lec 8 | MIT 3.091SC Introduction to Solid State Chemistry, Fall 2010
Lecture 8: Ionic Crystals; Born-Haber Cycle Instructor: Donald Sadoway View the complete course: http://ocw.mit.edu/3-091SCF10 License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
From playlist MIT 3.091SC Introduction to Solid State Chemistry, Fall 2010
Stirring the Mandelbrot Set: a checkerboard
http://code.google.com/p/mandelstir/
From playlist mandelstir
What Happened to Rudolf Hess' Plane?
Go to: https://curiositystream.thld.co/markfelton_0422 and use the code MARKFELTON to save 25% off today, that's only $14.99 a year. Thanks to Curiosity Stream for sponsoring today's video! The strange story of Rudolf Hess arriving in Scotland in 1941 remains unresolved even today. But wh
From playlist World War II Leaders
Rudolf Hess: The Last Prisoner of Spandau
Rudolf Hess, the last prisoner in Spandau, Berlin, died in 1987. He had been confined inside since 1947 with six other Nuremberg defendants, but by 1966 he was the last one left in a prison designed for 600. Find out how he ended up in Spandau, how he was guarded by the armed forces of fou
From playlist Hitler Various Subjects
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From playlist Royal Family Trees
The American Revolution - OverSimplified (Part 2)
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From playlist OverSimplified History
Farmer Hero Who Captured Rudolf Hess (1941) | Unissued Nº10 | British Pathé
This film was unused by British Pathé editors of the time and not screened in cinemas. In an attempt to bring hidden films to light, we have decided to create "British Pathé Unissued". Links to other videos related to this topic: Rudolf Hess Speaking (1930): https://www.britishpathe.com/
From playlist British Pathé Unissued
Enthalpy: Crash Course Chemistry #18
Energy is like the bestest best friend ever and yet, most of the time we take it for granted. Hank feels bad for our friend and wants us to learn more about it so that we can understand what it's trying to tell us - like that any bond between two atoms contains energy. How much energy? Tha
From playlist Chemistry
The King, the Thief, and the Spy - A Still Secret WW2 Scandal
Thanks to Established Titles for sponsoring today's video. Established Titles is now running a great Labor day Sale, plus get an additional 10% off on any purchase with code MARKFELTON. Go to https://establishedtitles.com/MARKFELTON and help support the channel. What does a former King of
From playlist The Cold War 1945-91
IB Chemistry Topic 5 Energetics 5.2 Hess's Law with enthalpy of formation and enthalpy of combustion
IB Chemistry Topic 5 Energetics 5.2 Hess's Law with enthalpy of formation and enthalpy of combustion How to use Hess' Law to work out enthalpy problems. The most valuable think to learn is how to work out if the formula is products less reactants or reactants less products. Practice proble
From playlist Topic 5/15 Energetics and thermochemistry
This Mandelbulb fractal is real!
A fractal just grew from my desk! Testing my Shadertoy plugin for After Effects ^_^
From playlist Shorts
Year 11 Chemistry Drivers of Reactions Applying Hess' Law
From playlist Y11 Chem Mod 4 Drivers of Reactions